Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 162 / 2 / iv

Table of contents
- Solution iv

Solution

Write

S (ζ) = \sum_{j \geq 0} ζ^{j}

. By the projective bundle formula, it is enough to verify the identity on a basis element

β = ζ^{q} \cap p^{*} α

,

0 \leq q \leq e

. Expand

p_{*} (S (ζ) \cap β) = \sum_{i \geq 0} p_{*} (ζ^{i + q} \cap p^{*} α)

and use the pushforward identities obtained from part iii. Multiplication by

c (p^{*} E) S (ζ)

is the inverse triangular operation, because

c (ξ) = c (p^{*} E) S (ζ) .

It recovers

ζ^{q} p^{*} α

k

. By linearity, every

β \in A_{k} (P (E))

satisfies

{c (p^{*} E) \cap \sum_{j \geq 0} ζ^{j} \cap p^{*} p_{*} (\sum_{i \geq 0} ζ^{i} \cap β)}_{k} = β .

Solved by gpt-5.6-sol high.

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